6. Find the order of every element in the symmetry group of the square, $$ D_{4} $$.

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1. Let the rotations be denoted by $$ r $$ and reflections by $$ s $$. The group $$ D_4 $$ has eight elements which can be written as:
   - $$ e $$ (the identity),
   - $$ r $$ (rotation by $$ 90^\circ $$),
   - $$ r^2 $$ (rotation by $$ 180^\circ $$),
   - $$ r^3 $$ (rotation by $$ 270^\circ $$),
   - $$ s $$ (a reflection),
   - $$ sr $$ (a reflection followed by a rotation by $$ 90^\circ $$),
   - $$ sr^2 $$ (a reflection followed by a rotation by $$ 180^\circ $$),
   - $$ sr^3 $$ (a reflection followed by a rotation by $$ 270^\circ $$).
   
2. To determine the order of each element:

- **Identity $$ e $$:**  
     The identity satisfies  
     $$
     e^1 = e.
     $$
     Therefore, the order of $$ e $$ is $$ 1 $$.

- **Rotation $$ r $$ by $$ 90^\circ $$:**  
     Since  
     $$
     r^4 = e,
     $$
     the order of $$ r $$ is $$ 4 $$.

- **Rotation $$ r^2 $$ by $$ 180^\circ $$:**  
     Notice that  
     $$
     (r^2)^2 = r^4 = e,
     $$
     so the order of $$ r^2 $$ is $$ 2 $$.

- **Rotation $$ r^3 $$ by $$ 270^\circ $$:**  
     Similarly,  
     $$
     (r^3)^4 = r^{12} = (r^4)^3 = e^3 = e,
     $$
     and the smallest power for which this happens is $$ 4 $$, so the order of $$ r^3 $$ is $$ 4 $$.

- **Reflections $$ s $$, $$ sr $$, $$ sr^2 $$, $$ sr^3 $$:**  
     For any reflection $$ x $$ in $$ D_4 $$, we have  
     $$
     x^2 = e.
     $$
     Thus, the order of each reflection $$ s $$, $$ sr $$, $$ sr^2 $$, and $$ sr^3 $$ is $$ 2 $$.

3. Summary of Orders:

- $$ o(e) = 1 $$
   - $$ o(r) = 4 $$
   - $$ o(r^2) = 2 $$
   - $$ o(r^3) = 4 $$
   - $$ o(s) = 2 $$
   - $$ o(sr) = 2 $$
   - $$ o(sr^2) = 2 $$
   - $$ o(sr^3) = 2 $$
The orders of the elements in the symmetry group $$ D_4 $$ of the square are as follows: - **Identity ($$ e $$)**: Order 1 - **Rotation by $$ 90^\circ $$ ($$ r $$)**: Order 4 - **Rotation by $$ 180^\circ $$ ($$ r^2 $$)**: Order 2 - **Rotation by $$ 270^\circ $$ ($$ r^3 $$)**: Order 4 - **Reflections ($$ s $$, $$ sr $$, $$ sr^2 $$, $$ sr^3 $$)**: Each has Order 2

Find the order of every element in the symmetry group of the square, .

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